We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in Lp(R3)for 1<p<3 and unbounded otherwise.

The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in $\mathbb{R}^3$, the perturbed Laplacian satisfies the same $L^p$−$L^q$ estimates of the free Laplacian in the smaller regime $q \in [2,3)$. These estimates are implied by a recent result concerning the Lpboundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime $q \geq 3$.

We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the ``projection counting'' scheme.

JF - Advances in Quantum Mechanics: Contemporary Trends and Open Problems
PB - Springer International Publishing
CY - Cham
SN - 978-3-319-58904-6
UR - https://doi.org/10.1007/978-3-319-58904-6_15
ER -
TY - RPRT
T1 - A class of Hamiltonians for a three-particle fermionic system at unitarity
Y1 - 2015
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
A1 - Domenico Finco
A1 - Alessandro Michelangeli
A1 - Alessandro Teta
AB - We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass $m$, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for $m$ larger than a critical value $m^* \simeq (13.607)^{-1}$ a self-adjoint and lower bounded Hamiltonian $H_0$ can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for $m\in(m^*,m^{**})$, where $m^{**}\simeq (8.62)^{-1}$, there is a further family of self-adjoint and lower bounded Hamiltonians $H_{0,\beta}$, $\beta \in \mathbb{R}$, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.
UR - http://urania.sissa.it/xmlui/handle/1963/34469
N1 - This SISSA preprint is composed of 29 pages and is recorded in PDF format
U1 - 34644
U2 - Mathematics
U4 - 1
ER -
TY - RPRT
T1 - Schödinger operators on half-line with shrinking potentials at the origin
Y1 - 2015
A1 - Gianfausto Dell'Antonio
A1 - Alessandro Michelangeli
AB - We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential.
PB - SISSA
UR - http://urania.sissa.it/xmlui/handle/1963/34439
U1 - 34566
ER -
TY - RPRT
T1 - Dynamics on a graph as the limit of the dynamics on a "fat graph"
Y1 - 2014
A1 - Gianfausto Dell'Antonio
A1 - Alessandro Michelangeli
AB - We discuss how the vertex boundary conditions for the dynamics of a quantum particle constrained on a graph emerge in the limit of the dynamics of a particle in a tubular region around the graph (\fat graph") when the transversal section of this region shrinks to zero. We give evidence of the fact that if the limit dynamics exists and is induced by the Laplacian on the graph with certain self-adjoint boundary conditions, such conditions are determined by the possible presence of a zero energy resonance on the fat graph. Pictorially, one may say that in the shrinking limit the resonance acts as a bridge connecting the boundary values at the vertex along the different rays.
PB - SISSA
UR - http://urania.sissa.it/xmlui/handle/1963/7485
U1 - 7598
ER -
TY - JOUR
T1 - Some remarks on quantum mechanics
JF - International Journal of Geometric Methods in Modern Physics, Volume 9, Issue 2, March 2012, Article number1260018
Y1 - 2012
A1 - Gianfausto Dell'Antonio
KW - Quantum mechanics
AB - We discuss the similarities and differences between the formalism of Hamiltonian Classical Mechanics and of Quantum Mechanics and exemplify the differences through an analysis of tracks in a cloud chamber.
PB - World Scientific Publishing
UR - http://hdl.handle.net/1963/7018
U1 - 7013
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions
JF - Rev. Math. Phys. 24 (2012), 1250017
Y1 - 2012
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
A1 - Domenico Finco
A1 - Alessandro Michelangeli
A1 - Alessandro Teta
AB - We study the stability problem for a non-relativistic quantum system in\\r\\ndimension three composed by $ N \\\\geq 2 $ identical fermions, with unit mass,\\r\\ninteracting with a different particle, with mass $ m $, via a zero-range\\r\\ninteraction of strength $ \\\\alpha \\\\in \\\\R $. We construct the corresponding\\r\\nrenormalised quadratic (or energy) form $ \\\\form $ and the so-called\\r\\nSkornyakov-Ter-Martirosyan symmetric extension $ H_{\\\\alpha} $, which is the\\r\\nnatural candidate as Hamiltonian of the system. We find a value of the mass $\\r\\nm^*(N) $ such that for $ m > m^*(N)$ the form $ \\\\form $ is closed and bounded from below. As a consequence, $ \\\\form $ defines a unique self-adjoint and bounded from below extension of $ H_{\\\\alpha}$ and therefore the system is stable. On the other hand, we also show that the form $ \\\\form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs.
PB - World Scientific
UR - http://hdl.handle.net/1963/6069
U1 - 5955
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -
TY - JOUR
T1 - On the number of eigenvalues of a model operator related to a system of three particles on lattices
JF - J. Phys. A 44 (2011) 315302
Y1 - 2011
A1 - Gianfausto Dell'Antonio
A1 - Zahriddin I. Muminov
A1 - Y.M. Shermatova
PB - IOP Publishing
UR - http://hdl.handle.net/1963/5496
U1 - 5340
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -
TY - JOUR
T1 - Effective Schroedinger dynamics on $ ε$-thin Dirichlet waveguides via Quantum Graphs I: star-shaped graphs
JF - J. Phys. A 43 (2010) 474014
Y1 - 2010
A1 - Gianfausto Dell'Antonio
A1 - Emanuele Costa
AB - We describe the boundary conditions at the vertex that one must choose to obtain a dynamical system that best describes the low-energy part of the evolution of a quantum system confined to a very small neighbourhood of a star-shaped metric graph.
PB - IOP Publishing
UR - http://hdl.handle.net/1963/4106
U1 - 298
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - A time-dependent perturbative analysis for a quantum particle in a cloud chamber
JF - Annales Henri Poincare 11 (2010) 539-564
Y1 - 2010
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
AB - We consider a simple model of a cloud chamber consisting of a test particle (the alpha-particle) interacting with two other particles (the atoms of the vapour) subject to attractive potentials centered in $a_1, a_2 \\\\in \\\\mathbb{R}^3$. At time zero the alpha-particle is described by an outgoing spherical wave centered in the origin and the atoms are in their ground state. We show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless $a_2$ lies on the line joining the origin with $a_1$. The work is a fully time-dependent version of the original analysis proposed by Mott in 1929.
PB - Springer
UR - http://hdl.handle.net/1963/3969
U1 - 432
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - The number of eigenvalues of three-particle Schrödinger operators on lattices
JF - J. Phys. A 40 (2007) 14819-14842
Y1 - 2007
A1 - Sergio Albeverio
A1 - Gianfausto Dell'Antonio
A1 - Saidakhmat N. Lakaev
AB - We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\\\\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schr\\\\\\\"{o}dinger operator $H_{\\\\gamma}(K),$ $K$ being the total quasi-momentum and $\\\\gamma>0$ the ratio of the mass of fermion and boson.\\nWe choose for $\\\\gamma>0$ the interaction $v(\\\\gamma)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance.\\nWe prove for any $\\\\gamma> 0$ the existence infinitely many eigenvalues of the operator $H_{\\\\gamma}(0).$ We establish for the number $N(0,\\\\gamma; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \\\\lim_{z\\\\to 0-}\\\\frac{N(0,\\\\gamma;z)}{\\\\mid \\\\log \\\\mid z\\\\mid \\\\mid}={U} (\\\\gamma) .$$ Moreover, for all nonzero values of the quasi-momentum $K \\\\in T^3 $ we establish the finiteness of the number $ N(K,\\\\gamma;\\\\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,\\\\gamma;0)$ of eigenvalues below zero.
UR - http://hdl.handle.net/1963/2576
U1 - 1545
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Decay of a bound state under a time-periodic perturbation: a toy case
JF - J. Phys. A 38 (2005) 4769-4781
Y1 - 2005
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
AB - We study the time evolution of a three dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with ``strength\\\'\\\' (\\\\alpha(t)). Under very weak generic conditions on the Fourier coefficients of (\\\\alpha(t)), we prove complete ionization as (t \\\\to \\\\infty). We prove also that, under the same conditions, all the states of the system are scattering states.
UR - http://hdl.handle.net/1963/2298
U1 - 1718
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Ionization for Three Dimensional Time-dependent Point Interactions
JF - Comm. Math. Phys. 257 (2005) 169-192
Y1 - 2005
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Andrea Mantile
AB - We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the ``strength\\\'\\\' of the interaction (\\\\alpha(t)) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of (\\\\alpha(t)), we prove that there is complete ionization as (t \\\\to \\\\infty), starting from a bound state at time (t = 0). Moreover we prove also that, under the same conditions, all the states of the system are scattering states.
UR - http://hdl.handle.net/1963/2297
U1 - 1719
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity
JF - Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004) 121-137
Y1 - 2004
A1 - Riccardo Adami
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
AB - We present some results on the blow-up phenomenon for the Schroedinger equation in dimension three with a nonlinear term supported in a fixed point. We find sufficient conditions for the blow up exploiting the moment of inertia of the solution and the uncertainty principle. In the critical case, we discuss the additional symmetry of the equation and construct a family of explicit blow up solutions.
PB - Elsevier
UR - http://hdl.handle.net/1963/2998
U1 - 1335
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Rotating Singular Perturbations of the Laplacian
JF - Ann. Henri Poincare 5 (2004) 773-808
Y1 - 2004
A1 - Michele Correggi
A1 - Gianfausto Dell'Antonio
AB - We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\\\\omega \\\\to \\\\infty).
PB - Springer
UR - http://hdl.handle.net/1963/2945
U1 - 1755
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Semiclassical analysis of constrained quantum systems
JF - J. Phys. A 37 (2004) 5605-5624
Y1 - 2004
A1 - Gianfausto Dell'Antonio
A1 - Lucattilio Tenuta
AB - We study the dynamics of a quantum particle in R^(n+m) constrained by a strong potential force to stay within a distance of order hbar (in suitable units) from a smooth n-dimensional submanifold M. We prove that in the semiclassical limit the evolution of the wave function is approximated in norm, up to terms of order hbar^(1/2), by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system.
PB - IOP Publishing
UR - http://hdl.handle.net/1963/2997
U1 - 1336
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Diffusion of a particle in presence of N moving point sources
JF - Annales Poincare Phys.Theor.69:413-424,1998
Y1 - 1998
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
PB - SISSA Library
UR - http://hdl.handle.net/1963/134
U1 - 75
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - JOUR
T1 - Statistics in space dimension two
JF - Lett. Math. Phys. 40 (1997), no. 3, 235-256
Y1 - 1997
A1 - Gianfausto Dell'Antonio
A1 - Rodolfo Figari
A1 - Alessandro Teta
AB - We construct as a selfadjoint operator the Schroedinger hamiltonian for a system of $N$ identical particles on a plane, obeying the statistics defined by a representation $\\\\pi_1$ of the braid group. We use quadratic forms and potential theory, and give details only for the free case; standard arguments provide the extension of our approach to the case of potentials which are small in the sense of forms with respect to the laplacian. We also comment on the relation between the analysis given here and other approaches to the problem, and also on the connection with the description of a quantum particle on a plane under the influence of a shielded magnetic field (Aharanov-Bohm effect).
PB - SISSA Library
UR - http://hdl.handle.net/1963/130
U1 - 12
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - JOUR
T1 - Classical solutions for a perturbed N-body system
JF - Topological nonlinear analysis, II (Frascati, 1995), 1--86, Progr. Nonlinear Differential Equations Appl., 27, Birkhauser Boston, Boston, MA, 1997
Y1 - 1995
A1 - Gianfausto Dell'Antonio
PB - SISSA Library
UR - http://hdl.handle.net/1963/126
U1 - 16
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - JOUR
T1 - Workshop on point interactions, Trieste, 21-23 December 1992
Y1 - 1993
A1 - Gianfausto Dell'Antonio
PB - SISSA Library
UR - http://hdl.handle.net/1963/71
U1 - 52
U2 - LISNU
U3 - Interdisciplinary Laboratory for Advanced Studies
ER -
TY - JOUR
T1 - On the number of families of periodic solutions of a Hamiltonian system near equilibrium. II. (English. Italian summary)
JF - Boll. Un. Mat. Ital. B (7) 3 (1989), no. 3, 579-590
Y1 - 1989
A1 - Gianfausto Dell'Antonio
A1 - Biancamaria D'Onofrio
PB - SISSA Library
UR - http://hdl.handle.net/1963/609
U1 - 3295
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Methods of stochastic stability and properties of the Gribov horizon in the stochastic quantization of gauge theories
JF - Stochastic processes, physics and geompetry (Ascona and Locarno, 1988), 302, World Sci.Publishing,NJ(1990)
Y1 - 1988
A1 - Gianfausto Dell'Antonio
PB - SISSA Library
UR - http://hdl.handle.net/1963/817
U1 - 2974
U2 - Mathematics
U3 - Mathematical Physics
ER -